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  <h1 id="数学-高等数学 第8-9讲 一元函数积分学" class="content-subhead">数学-高等数学 第8-9讲 一元函数积分学</h1>
  <p>
    <span>1970-01-01</span>
    <span><span class="post-category post-category-math">Math</span></span>
    <span id="/public/article/数学-高等数学 4 第8-9讲 一元函数积分学.html" class="leancloud_visitors" style="display:none" data-flag-title="数学-高等数学 第8-9讲 一元函数积分学"></span>
  </p>
  <h2 id="8-9">第8-9讲 一元函数积分学</h2>
<h3 id="1">1. 公式</h3>
<h4 id="1_1">（1）重要的公式</h4>
<p>
<script type="math/tex; mode=display">
\begin{equation}\begin{split}
                                 \int a^xdx &= \cfrac{a^x}{\ln a} + C \\[2em]
        (*\ x=a\tan t)\ \ \ \ \int\cfrac{1}{x^2+a^2}dx &= \cfrac{1}{a}\arctan\cfrac{x}{a} + C               & (a\gt0) \\[1ex]
        (*)\ \ \ \ \int\cfrac{1}{x^2-a^2}dx &= \cfrac{1}{2a}\ln\bigg|\cfrac{x-a}{x+a}\bigg| + C  & (a\gt0) \\[1ex]
       (*)\ \ \ \ \int\cfrac{1}{-x^2+a^2}dx &= \cfrac{1}{2a}\ln\bigg|\cfrac{x+a}{-x+a}\bigg| + C & (a\gt0) \\[2em]
 (*)\ \ \ \ \int\cfrac{1}{\sqrt{x^2+a^2}}dx &= \ln\bigg|x+\sqrt{x^2+a^2}\bigg| + C \\[1ex]
 (*)\ \ \ \ \int\cfrac{1}{\sqrt{x^2-a^2}}dx &= \ln\bigg|x+\sqrt{x^2-a^2}\bigg| + C &\ \ \ \ (|x|\gt|a|) \\[1ex]
(*\ x=a\sin t)\ \ \ \ \int\cfrac{1}{\sqrt{-x^2+a^2}}dx &= \arcsin\cfrac{x}{a} + C             & (a\gt0) \\[1ex]
                                            &=-\arccos\cfrac{x}{a} + C             & (a\gt0)
\end{split}\end{equation}
</script>
</p>
<h4 id="2">（2）被积函数包含三角函数</h4>
<p>
<script type="math/tex; mode=display">
\begin{equation}\begin{split}
     \int\csc xdx = \int\cfrac{1}{\sin x}dx &=-\ln\bigg|\csc x + \cot x\bigg| + C \\[1ex]
     \int\sec xdx = \int\cfrac{1}{\cos x}dx &= \ln\bigg|\sec x + \tan x\bigg| + C \\[2em]
                              \int\tan  xdx &=-\ln\bigg|\cos x\bigg| + C \\[1ex]
     \int\cot xdx = \int\cfrac{1}{\tan x}dx &= \ln\bigg|\sin x\bigg| + C \\[2em]
                              \int\sin^2xdx &= \cfrac{x}{2} - \cfrac{\sin 2x}{4} + C \\[1ex]
                              \int\cos^2xdx &= \cfrac{x}{2} + \cfrac{\sin 2x}{4} + C \\[1ex]
     \int\csc^2x = \int\cfrac{1}{\sin^2}xdx &=-\cot x + C \\[1ex]
     \int\sec^2x = \int\cfrac{1}{\cos^2}xdx &= \tan x + C \\[2em]
                              \int\tan^2xdx &= \tan x - x + C \\[1ex]
     \int\cot^2x = \int\cfrac{1}{\tan^2x}dx &=-\cot x - x + C \\[2em]
\int\csc x\cot xdx = \int\cfrac{\cos x}{\sin^2x}dx &=-\csc x + C \\[1ex]
\int\sec x\tan xdx = \int\cfrac{\sin x}{\cos^2x}dx &= \sec x + C \\[2em]
\end{split}\end{equation}
</script>
</p>
<h3 id="2_1">2. 不定积分的计算方法</h3>
<h4 id="1_2">（1）凑微</h4>
<h4 id="2_2">（2）换元</h4>
<h4 id="3">（3）分部积分</h4>
<p>
<script type="math/tex; mode=display">
\begin{equation}\begin{split}
\int u(x)v'(x)dx &= \int u(x)dv(x) \\
&= u(x)v(x) - \int u'(x)v(x)dx
\end{split}\end{equation}
</script>
</p>
<h3 id="3_1">3. 定积分的计算</h3>
<h4 id="1_3">（1）定积分的定义</h4>
<h5 id="1_4">1、均匀分割</h5>
<p>
<script type="math/tex; mode=display">
\begin{equation}\begin{split}
区间\ [0,1]\ 为&\ \bigg\{\cfrac{1}{n},\ \cfrac{1}{n},\ \cfrac{1}{n}\cdots,\ \cfrac{1}{n}\bigg\} \\[2ex]
[x_{k-1},x_{k}]&=[\cfrac{k-1}{n},\cfrac{k}{n}] \\[2ex]
每段\ x_{k}-x_{k-1}&=\cfrac{1}{n} \\[2ex]
&\xi_k为区间[x_{k-1},x_{k}]即[\cfrac{k-1}{n},\cfrac{k}{n}]的端点 \\[2ex]
(1)\ \ \ \ \ \ \ \ \ \ \int_o^1{f(x)dx}
&=\lim_{n\to\infty}\sum_{k=1}^n{f(\xi_k)\cfrac{1}{n}} 
=\lim_{n\to\infty}\sum_{k=1}^n{f(\cfrac{k}{n})\cfrac{1}{n}} \\[2ex]
(2)\ \ \ \ \ \ \ \ \ \ \int_o^1{f(x)dx}
&=\lim_{n\to\infty}\sum_{k=1}^n{f(\xi_k)\cfrac{1}{n}} 
=\lim_{n\to\infty}\sum_{k=1}^n{f(\cfrac{k-1}{n})\cfrac{1}{n}} \\[3em]
&\xi_k为区间[x_{k-1},x_{k}]即[\cfrac{k-1}{n},\cfrac{k}{n}]的中点 \\[1ex]
&\xi_k=\cfrac{x_{k-1}+x_{k}}{2}=\cfrac{2k-1}{2n} \\[2ex]
(3)\ \ \ \ \ \ \ \ \ \ \int_o^1{f(x)dx}
&=\lim_{n\to\infty}\sum_{k=1}^n{f(\xi_k)\cfrac{1}{n}} 
=\lim_{n\to\infty}\sum_{k=1}^n{f(\cfrac{2k-1}{2n})\cfrac{1}{n}} \\[3em]
&\xi_k为区间[x_{k-1},x_{k}]即[\cfrac{k-1}{n},\cfrac{k}{n}]的几何平均 \\[1ex]
&\xi_k=\sqrt{x_{k-1}x_{k}}=\cfrac{\sqrt{(k-1)k}}{n} \\[2ex]
(4)\ \ \ \ \ \ \ \ \ \ \int_o^1{f(x)dx}
&=\lim_{n\to\infty}\sum_{k=1}^n{f(\xi_k)\cfrac{1}{n}} 
=\lim_{n\to\infty}\sum_{k=1}^n{f(\cfrac{\sqrt{(k-1)k}}{n})\cfrac{1}{n}}
\end{split}\end{equation}
</script>
</p>
<h5 id="2_3">2、等差分割</h5>
<p>
<script type="math/tex; mode=display">
\begin{equation}\begin{split}
区间\ [0,1]\ 为&\ \bigg\{1l,\ 2l,\ 3l,\ \cdots,\ (k-1)l,\ kl,\ \cdots,\ nl\bigg\} \\[2ex]
[x_{k-1},x_{k}]
=&[\cfrac{\cfrac{(k-1)k}{2}l}{\cfrac{n(1+n)}{2}l},\cfrac{\cfrac{k(1+k)}{2}l}{\cfrac{n(1+n)}{2}l}] \\[2ex]
&=[\cfrac{(k-1)k}{n(1+n)},\cfrac{k(1+k)}{n(1+n)}] \\[3ex]
每段\ x_{k}-x_{k-1}&=\cfrac{2k}{n(1+n)}
\end{split}\end{equation}
</script>
</p>
<h5 id="3_2">3、等比分割</h5>
<p>
<script type="math/tex; mode=display">
\begin{equation}\begin{split}
区间\ [0,1]\ 为&\ \bigg\{2^0l,\ 2^1l,\ 2^2l,\ \cdots,\ 2^{k-2}l,\ 2^{k-1}l, \ \cdots\ 2^{n-1}l\bigg\} \\[2ex]
[x_{k-1},x_{k}]
&=[\cfrac{\cfrac{2^0(1-2^{k-1})}{1-2}l}{\cfrac{2^0(1-2^n)}{1-2}l},
\cfrac{\cfrac{2^0(1-2^k)}{1-2}l}{\cfrac{2^0(1-2^n)}{1-2}l}] \\[2ex]
&=[\cfrac{2^{k-1}-1}{2^n-1},
\cfrac{2^k-1}{2^n-1}] \\[3ex]
每段\ x_{k}-x_{k-1}
&=\cfrac{2^k-2^{k-1}}{2^n-1} \\[2ex]
&=\cfrac{2^{k-1}}{2^n-1}
\end{split}\end{equation}
</script>
</p>
<h4 id="2_4">（2）重要公式</h4>
<p>
<script type="math/tex; mode=display">
\begin{equation}\begin{split}
(区间再现公式)\int_a^b f(x)dx &= \int_a^b f(a+b-x)dx \\[1ex]
                &= \cfrac{1}{2}\int_a^b \bigg[f(x) + f(a+b-x)\bigg]dx \\[1ex]
                &= \int_a^{\frac{a+b}{2}} \bigg[f(x) + f(a+b-x)\bigg]dx \\[2em]
(区间简化公式)\ \ \ x-\cfrac{a+b}{2}&=\cfrac{b-a}{2}\sin t \\[1ex]
\int_a^b f(x)dx &= \int_{\frac{\pi}{2}}^{\frac{\pi}{2}}\bigg[f(\cfrac{a+b}{2}+\cfrac{b-a}{2}\sin t)\cfrac{b-a}{2}\cos t\bigg]dt \\[2em]
(区间简化公式)\ \ \ x-a&=(b-a)t \\[1ex]
\int_a^b f(x)dx &= \int_0^1 (b-a)f[a+(b-a)t]dt \\[2em]
\int_{-a}^af(x)dx &= \int_0^a[f(x)+f(-x)]dx \ \ \ \ \ \ (a>0)
\end{split}\end{equation}
</script>
</p>
<h4 id="3_3">（3）华式公式（“点火公式”）</h4>
<p>
<script type="math/tex; mode=display">
\begin{equation}\begin{split}
\int_0^{\frac{\pi}{2}}\sin^nx\ dx &= \\[1ex]
\int_0^{\frac{\pi}{2}}\cos^nx\ dx &= 
\begin{cases}
\cfrac{n-1}{n}·\cfrac{n-3}{n-2}\cdots\cfrac{2}{3} &n为大于1的奇数  \\[2ex]
\cfrac{n-1}{n}·\cfrac{n-3}{n-2}\cdots\cfrac{1}{2}·\cfrac{\pi}{2} &n为正偶数
\end{cases}  \\[2em]
\int_0^{\pi}\sin^nx\ dx &= 
\begin{cases}
2\int_0^{\frac{\pi}{2}}\sin^nx\ dx &n为正奇数  \\[1ex]
2\int_0^{\frac{\pi}{2}}\sin^nx\ dx &n为正偶数
\end{cases} \\[1ex]
\int_0^{\pi}\cos^nx\ dx &=
\begin{cases}
0                                  &n为正奇数  \\[1ex]
2\int_0^{\frac{\pi}{2}}\cos^nx\ dx &n为正偶数
\end{cases}  \\[2em]
\int_0^{2\pi}\sin^nx\ dx &=
\begin{cases}
0                                  &n为正奇数  \\[1ex]
4\int_0^{\frac{\pi}{2}}\sin^nx\ dx &n为正偶数
\end{cases}  \\[1ex]
\int_0^{2\pi}\cos^nx\ dx &=
\begin{cases}
0                                  &n为正奇数  \\[1ex]
4\int_0^{\frac{\pi}{2}}\cos^nx\ dx &n为正偶数
\end{cases}  \\[2em]
\int_0^{\pi}xf(\sin x)dx &= \cfrac{\pi}{2}\int_0^{\pi}f(\sin x)dx = \pi\int_0^{\frac{\pi}{2}}xf(\sin x)dx \\[1ex]
\int_0^{nT}xf(x)dx &= \cfrac{n^2 T}{2}\int_0^T f(x)dx\\[2em]            
\int_0^{\frac{\pi}{2}}f(\sin x)dx &= \int_0^{\frac{\pi}{2}}f(\cos x)dx \\[1ex]
\int_0^{\frac{\pi}{2}}f(\sin x, \cos x)dx &= \int_0^{\frac{\pi}{2}}f(\cos x, \sin x)dx \\[2em]
\end{split}\end{equation}
</script>
</p>
<h4 id="4">（4）伽马函数</h4>
<p>
<script type="math/tex; mode=display">
实数域：\Gamma(x) = \int_0^{+\infty}t^{x-1}e^{-t}dt,\ (x>0)
</script>
</p>
<p>伽马函数的推导<br />
<script type="math/tex; mode=display">
\begin{equation}\begin{split}
\cfrac{1}{1-x} &= \sum_{n=0}^\infty x^n (对比)\\[1ex]
&= \int_0^{+\infty}e^{-(1-x)t}dt \\[1ex]
&= \int_0^{+\infty}e^{-t+xt}dt \\[1ex]
&= \int_0^{+\infty}e^{-t}\sum_{n=0}^\infty \cfrac{(xt)^n}{n!}dt \\[1ex]
&= \sum_{n=0}^\infty \cfrac{\int_0^{+\infty}t^ne^{-t} dt}{n!}x^n (对比)\\[2em]
\int_0^{+\infty}t^ne^{-t} dt &= n!\ \ \ \ 【便利公式】
\end{split}\end{equation}
</script>
</p>
<h3 id="4_1">4. 变限积分的计算</h3>
<p>
<script type="math/tex; mode=display">
\begin{equation}\begin{split}
\bigg[\int_a^{\varphi(x)}f(t)dt\bigg]_x' &= f[\varphi(x)]·\varphi'(x) \\[1ex]
\bigg[\int_{\varphi_1(x)}^{\varphi_2(x)}f(t)dt\bigg]_x' &= f[\varphi_2(x)]·\varphi_2'(x) - f[\varphi_1(x)]·\varphi_1'(x)
\end{split}\end{equation}
</script>
</p>
<h3 id="5">5. 反常积分的计算</h3>
<p><img class="pure-img" src="https://zromyk.gitee.io/myblog-figurebed/post/数学-高等数学.assets/1203675-20171207172202738-17246288.png" alt="1203675-20171207172202738-17246288" style="zoom:67%;" /></p>
<p><img class="pure-img" src="https://zromyk.gitee.io/myblog-figurebed/post/数学-高等数学.assets/1203675-20171207172925972-145324329.png" alt="1203675-20171207172925972-145324329" style="zoom:67%;" /><br />
<script type="math/tex; mode=display">
\begin{equation}\begin{split}
\int_a^{+\infty}f(x)dx &= \lim_{x\to+\infty}F(x)-F(a) \\[1em]
(a为瑕点)\int_a^b f(x)dx &= F(b)-\lim_{x\to a}F(x)
\end{split}\end{equation}
</script>
</p>
<p>
<script type="math/tex; mode=display">
\begin{equation}\begin{split}
p\ 积分：
\int_a^{+\infty}\cfrac{1}{x^p}dx
&\begin{cases}
\text{收敛}, & q\gt1 \\[2ex]
\text{发散}, & q\le1
\end{cases} \\[2em]
广义\ p\ 积分：
\int_a^{+\infty}\cfrac{1}{x^\alpha(\ln x)^\beta}dx
&\begin{cases}
\text{收敛}, &\alpha\gt1\ 或\ \alpha=1,\beta\gt1 \\[2ex]
\text{发散}, &\alpha\lt1\ 或\ \alpha=1,\beta\le1 \\[2ex]
\end{cases} \\[2em]
瑕积分：
\int_0^1\cfrac{1}{x^p}dx
&\begin{cases}
\text{收敛}, & 【\ q\lt1\ 】 \\[2ex]
\text{发散}, & 【\ q\ge1\ 】
\end{cases}
\end{split}\end{equation}
</script>
</p>
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  <a class="pure-menu-link" style="padding:0.1em 0em 0.1em 0.75em;" href="#8-9">第8-9讲 一元函数积分学</a>
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  <a class="pure-menu-link" style="padding:0.1em 0em 0.1em 1.25em;" href="#1">1. 公式</a>
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  <a class="pure-menu-link" style="padding:0.1em 0em 0.1em 1.75em;" href="#1_1">（1）重要的公式</a>
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  <a class="pure-menu-link" style="padding:0.1em 0em 0.1em 1.75em;" href="#2">（2）被积函数包含三角函数</a>
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  <a class="pure-menu-link" style="padding:0.1em 0em 0.1em 1.25em;" href="#2_1">2. 不定积分的计算方法</a>
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  <a class="pure-menu-link" style="padding:0.1em 0em 0.1em 1.75em;" href="#1_2">（1）凑微</a>
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  <a class="pure-menu-link" style="padding:0.1em 0em 0.1em 1.75em;" href="#2_2">（2）换元</a>
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  <a class="pure-menu-link" style="padding:0.1em 0em 0.1em 1.75em;" href="#3">（3）分部积分</a>
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  <a class="pure-menu-link" style="padding:0.1em 0em 0.1em 1.25em;" href="#3_1">3. 定积分的计算</a>
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  <a class="pure-menu-link" style="padding:0.1em 0em 0.1em 1.75em;" href="#1_3">（1）定积分的定义</a>
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  <a class="pure-menu-link" style="padding:0.1em 0em 0.1em 2.25em;" href="#1_4">1、均匀分割</a>
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  <a class="pure-menu-link" style="padding:0.1em 0em 0.1em 2.25em;" href="#2_3">2、等差分割</a>
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  <a class="pure-menu-link" style="padding:0.1em 0em 0.1em 2.25em;" href="#3_2">3、等比分割</a>
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  <a class="pure-menu-link" style="padding:0.1em 0em 0.1em 1.75em;" href="#2_4">（2）重要公式</a>
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  <a class="pure-menu-link" style="padding:0.1em 0em 0.1em 1.75em;" href="#3_3">（3）华式公式（“点火公式”）</a>
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  <a class="pure-menu-link" style="padding:0.1em 0em 0.1em 1.75em;" href="#4">（4）伽马函数</a>
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  <a class="pure-menu-link" style="padding:0.1em 0em 0.1em 1.25em;" href="#4_1">4. 变限积分的计算</a>
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  <a class="pure-menu-link" style="padding:0.1em 0em 0.1em 1.25em;" href="#5">5. 反常积分的计算</a>
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